Class Of Nonlinear Descriptor Systems Research Articles

State and adaptive disturbance observer co‐design for incrementally quadratic nonlinear descriptor systems with nonlinear outputs

AbstractThe problem of designing state and adaptive disturbance observers for a class of nonlinear descriptor systems with the disturbance generated by an unknown exogenous system is addressed in this paper. Especially, the nonlinear terms satisfy incremental quadratic constraints parameterized by a bunch of multiplier matrices, which can represent various classes of nonlinearities. For three different cases in the output equation, the state and adaptive disturbance observers are proposed. Some sufficient conditions formulated in terms of linear matrix inequalities and algebraic constraints are derived to guarantee that state and disturbance error systems, simultaneously, are asymptotically stable with performance level. The availability of proposed observers is validated by two practical examples.

Adaptive observer design for a class of nonlinear descriptor systems

This paper concerns the adaptive observer design for a class of nonlinear descriptor systems. It allows the simultaneous estimation of the system states and its parameters. The presented observer is in a general form, generalizing the design for the full order and reduced order observers. The existence conditions for the observer are given and the parameterization of all the observer parameters is provided. The design is obtained in the LMI formulation. The case of the linear descriptor system can be directly obtained from the presented results as a particular case. A numerical example is provided to illustrate the design procedure.

Input-output finite-time stability of nonlinear descriptor systems

This article discusses the input-output finite-time stability and stabilization problems of a class of nonlinear descriptor systems. Two classes of disturbance inputs which belong to [Formula: see text] and [Formula: see text] are considered. Under the impulse-free condition of the descriptor systems, the article first studies the input-output finite-time stability with respect to the nonlinear descriptor systems with [Formula: see text] and [Formula: see text] disturbance inputs. Based on this, the design of the state feedback controllers ensures that the closed-loop descriptor systems are input-output finite-time stabilization. Finally, the results of simulation on a nonlinear descriptor circuit system verify the effectiveness of the controllers in this article.

Admissibility and admissibilization for T–S fuzzy descriptor systems with partially unknown membership functions

In this paper, the admissibility analysis and controller design issues for a class of nonlinear descriptor systems described by the T–S descriptor fuzzy model with partially unknown membership functions are discussed. New sufficient conditions are first given to ensure the admissibility of such fuzzy descriptor systems. To fully utilise the information of membership functions, the known and unknown parts of membership functions are handled by staircase functions and the bounds of the unknown part, respectively. Different from the traditional parallel distribution compensation (PDC) method, the state-feedback controller design condition is given under an imperfect premise matching method via linear matrix inequalities (LMIs). Some examples are given to show the validity of the proposed methods.

Unknown input estimation algorithms for a class of LPV/nonlinear systems with application to wastewater treatment process

This paper addresses the problem of unknown input estimation for a class of nonlinear systems with mixed nonlinear terms, namely Linear Parameter Varying (LPV) parts and purely Lipschitz nonlinearities. Three new unknown input estimation algorithms are proposed, where each algorithm depends on the distribution of the unknown inputs in the system. These algorithms provide estimation of the maximum possible unknown inputs in a system, contrarily to the methods available in the literature, which consider only particular cases. Before introducing these estimation algorithms, a general LMI-based [Formula: see text] observer design methodology is provided, as a preliminary result, for a class of nonlinear descriptor systems with nonlinear outputs. To this end, a specific Lyapunov function is exploited to avoid derivatives of the disturbances. The proposed LMI conditions are less conservative than those existing in the literature. This is due to the specific Lyapunov function, the use of Young inequality in a judicious way, and the reformulation of the Lipschitz inequality. The proposed algorithms are applied to a wastewater treatment model to show their effectiveness and performances.

Emerging approaches for nonlinear parameter varying systems

Emerging approaches for nonlinear parameter varying systems

Takagi–Sugeno fuzzy observer design for nonlinear descriptor systems with unmeasured premise variables and unknown inputs

AbstractThis article presents a new observer design framework for a class of nonlinear descriptor systems with unknown but bounded inputs. In the presence of unmeasured nonlinearities, that is, premise variables, designing nonlinear observers is known as particularly challenging. To solve this problem, we rewrite the nonlinear descriptor system in the form of a Takagi–Sugeno (TS) fuzzy model with nonlinear consequents. This model reformulation enables an effective use of the differential mean value theorem to deal with the mismatching terms involved in the estimation error dynamics. These nonlinear terms, issued from the unmeasured nonlinearities of the descriptor system, cause a major technical difficulty for TS fuzzy‐model‐based observer design. The descriptor form is treated through a singular redundancy representation. For observer design, we introduce into the Luenberger‐like observer structure a virtual variable aiming at estimating the one‐step ahead state. This variable introduction allows for free‐structure decision variables involved in the observer design to further reduce the conservatism. Using Lyapunov‐based arguments, the observer design is reformulated as an optimization problem under linear matrix inequalities with a single line search parameter. The estimation error bounds of both the state and the unknown input can be minimized by means of a guaranteed ℓ∞‐gain performance level. The interests of the new ℓ∞ TS fuzzy observer design are clearly illustrated with two physically motivated examples.

A Nonlinear Observer for Robust Fault Reconstruction in One-Sided Lipschitz and Quadratically Inner-Bounded Nonlinear Descriptor Systems

This article introduces a nonlinear observer to reconstruct faults in a class of nonlinear descriptor systems affected by disturbances, in both the state and output equations. The fault enters the system through nonlinear functions in both its state and output equations. The original system is first transformed such that design freedom in its structure is easier to exploit, and then a nonlinear observer is designed based on the transformed system to reconstruct the fault. Linear matrix inequalities are used to determine the gains of the observer such that the effect of the disturbances on the fault reconstruction is bounded. Finally, three simulation examples are carried out to verify the effectiveness of the scheme.

Designing Fuzzy Descriptor Observer with Unmeasured Premise Variables for Head-Two-Arms-Trunk System

Abstract Using the technique of unknown input observer, this paper aims at estimating internal variables of people living with a complete spinal cord injury (SCI). The goal is to provide a better understanding on the sitting control strategy of SCI people. The observer design is based on a Head-Two-Arms-Trunk (H2AT) model, belonging to a class of nonlinear descriptor systems. For observer design, this model is represented in a specific Takagi-Sugeno (TS) fuzzy form with nonlinear consequents. In contrast to previous fuzzy estimation results based on conventional TS fuzzy modeling, the new TS formulation allows separating all unmeasured premise variables in the nonlinear consequent parts. This contributes to reduce the computational burden of the observer design and the structural complexity of the designed fuzzy observer. In particular, the new formulation enables a more effective way to deal with unmeasured premise variables. Using Lyapunov stability theorem, sufficient conditions to design the unknown input observer are derived in the form of linear matrix inequalities, conveniently solved by convex optimization techniques. Simulation results demonstrate the effectiveness of the proposed observer design.

Finite‐time stabilisation for a class of nonlinear descriptor systems

The finite-time stabilisation, finite-time H ∞ control and finite-time adaptive H ∞ control problems for a class of nonlinear descriptor systems are studied. First, under a sufficient condition for the system impulse controllable, a finite-time stabilised controller is designed by a suitable state feedback. Then, the finite-time H ∞ control and finite-time adaptive H ∞ control problems are considered for the nonlinear descriptor system with external disturbances and/or parametric perturbations, respectively. Compared with the existing results, there is no constraint for singular matrix E, therefore controllers can be designed for more nonlinear descriptor systems. Finally, an example of the nonlinear descriptor circuit system with simulations is applied to illustrate the effectiveness of the finite-time controllers obtained in this study.

Adaptive Parallel Simultaneous Stabilization of a Class of Nonlinear Descriptor Systems via Dissipative Matrix Method

This paper investigates the adaptive parallel simultaneous stabilization and robust adaptive parallel simultaneous stabilization problems of a class of nonlinear descriptor systems via dissipative matrix method. Firstly, under an output feedback law, two nonlinear descriptor systems are transformed into two nonlinear differential-algebraic systems by nonsingular transformations, and a sufficient condition of impulse-free is given for two resulting closed-loop systems. Then, the two systems are combined to generate an augmented dissipative Hamiltonian differential-algebraic system by using the system-augmentation technique. Based on the dissipative system, an adaptive parallel simultaneous stabilization controller and a robust adaptive parallel simultaneous stabilization controller are designed for the two systems. Furthermore, the case of more than two nonlinear descriptor systems is investigated. Finally, an illustrative example is studied by using the results proposed in this paper, and simulations show that the adaptive parallel simultaneous stabilization controllers obtained in this paper work very well.

Unknown input observer design for a class of nonlinear descriptor systems: a Takagi-Sugeno approach with Lipschitz constraints

Unknown input observer design for a class of nonlinear descriptor systems: a Takagi-Sugeno approach with Lipschitz constraints

Observer design for one-sided Lipschitz descriptor systems

This paper describes the design of a nonlinear observer for a class of nonlinear descriptor systems with one-sided Lipschitz nonlinearities. To provide a general framework applicable to a large class of systems, nonlinearities and disturbances are considered at the state as well as at the output equations. Non-strict and strict bilinear matrix inequality (BMI)-based robust observer synthesis schemes are provided by utilizing the one-sided Lipschitz condition, the concept of quadratic inner boundedness, the generalized Lyapunov theory for singular systems, and the L2 gain reduction. The BMI-based condition is converted into the linear matrix inequality (LMI)-based condition by utilizing change of variables for straightforward computation of the observer gain matrix. In contrast to the traditional observer schemes for one-sided Lipschitz nonlinear systems, the resultant scheme is applicable to singular systems, capable of dealing with nonlinearities at the output equation and appropriate for treating with disturbances. Two simulation examples for the one-sided Lipschitz and the singular one-sided Lipschitz nonlinear systems are provided to validate the proposed observer synthesis methodologies.

Observer Design for a Class of Nonlinear Descriptor Systems: A Takagi-Sugeno Approach with Unmeasurable Premise Variables

The Takagi-Sugeno (T-S) fuzzy observer for dynamical systems described by ordinary differential equations is widely discussed in the literature. The aim of this paper is to extend this observer design to a class of T-S descriptor systems with unmeasurable premise variables. In practice, the computation of solutions of differential-algebraic equations requires the combination of an ordinary differential equations (ODE) routine together with an optimization algorithm. Therefore, a natural way permitting to estimate the state of such a system is to design a procedure based on a similar numerical algorithm. Beside some numerical difficulties, the drawback of such a method lies in the fact that it is not easy to establish a rigorous proof of the convergence of the observer. The main result of this paper consists in showing that the state estimation problem for a class of T-S descriptor systems can be achieved by using a fuzzy observer having only an ODE structure. The convergence of the state estimation error is studied using the Lyapunov theory and the stability conditions are given in terms of linear matrix inequalities (LMIs). Finally, an application to a model of a heat exchanger pilot process is given to illustrate the performance of the proposed observer.

Dissipative control for a class of nonlinear descriptor systems

This paper is concerned with the dissipative control problem for a class of nonlinear descriptor systems. Based on Lyapunov stability theory, sufficient conditions are derived, which guarantee that the underlying systems are strictly dissipative. Then, the design method for a state feedback controller is provided. All the conditions can be expressed via linear matrix inequalities. Finally, a numerical example is presented to demonstrate the validity of the proposed methods.

An explicit fuzzy observer design for a class of Takagi-Sugeno descriptor systems

In [1] the authors proposed a study concerning the design of explicit state observer for a class of nonlinear descriptor systems described by Takagi-Sugeno (T-S) model with measurable premise variables. In this paper, the aim is to extend this result to a class of T-S descriptor systems when the premise variables are not measurables. This new approach is based on the singular value decomposition. The convergence of the state estimation error is studied using the Lyapunov theory and the stability conditions are given in terms of Linear Matrix Inequalities (LMIs). Finally, numerical simulations are given in order to highlight the performance of the proposed estimator.

Observer design for a class of nonlinear descriptor systems

The observer design problem for a class of nonlinear descriptor systems is considered. Nonlinear generalized proportional integral observers and nonlinear generalized proportional integral derivative observers are proposed. Using the differential mean value theorem, we transform the nonlinear error dynamics descriptor system into a linear parameter varying descriptor system. Then the stability of the error system is analyzed by a Lyapunov function, which shows that the errors are exponentially convergent. If the given linear matrix inequalities are feasible, all the gain matrices can be obtained. Compared with the existing observer design methods, the proposed methods offer more degrees of design freedom. As a result, a more comprehensive type of nonlinear descriptor systems can be designed and better transient performance and robustness against noise can be achieved. Finally, examples are given to illustrate the proposed methods.

ℋ −/ℋ ∞ fault detection filter for a class of nonlinear descriptor systems

This article addresses the problem of designing a robust fault detection filter (RFDF) for a class of nonlinear descriptor systems whose nonlinear term is globally Lipschitz and linear term is described by a linear parameter varying form. The effect of perturbations on the residuals is minimised using the ℋ∞ norm while the sensitivity performance is guaranteed using the ℋ− index. Thanks to a particular Lyapunov function: which depends on the faults and the filter design problem is formulated as a convex optimisation problem solved via linear matrix inequality techniques. Performances of the proposed filter are shown through the application of an activated sludge process model.

Actuator fault estimation for a class of nonlinear descriptor systems

This article proposes an actuator fault estimation approach for a class of nonlinear descriptor systems. The radial basis function (RBF) neural networks are utilised to model the actuator faults. The adaptive fault estimation observer is designed by exploiting the on-line learning ability of RBF neural networks to approximate the actuator fault. The adaptive algorithm of the RBF networks is established by the Lyapunov theory, and the design of the proposed observer is reformulated as a set of linear matrix inequalities (LMIs), which can be conveniently solved by standard LMI tools. Finally, two simulation examples are used to demonstrate the effectiveness of the proposed fault diagnosis method.

Input‐to‐state stability of a class of descriptor systems

SUMMARYThis paper studies the input‐to‐state stability (ISS) of descriptor systems with exogenous disturbances. on the basis of the ISS theory of standard state‐space nonlinear systems, a sufficient condition for a class of nonlinear descriptor system to be ISS is proved. Furthermore, a design method of the state feedback controllers is given to make the closed‐loop system ISS. A numerical example is given to illustrate the effectiveness of the controller design.Copyright © 2012 John Wiley & Sons, Ltd.